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User blog:P進大好きbot/Criterion of the Order-Preserving Property of FGH
I show that "Syst3ms' condition" implies the oreder-preserving property of FGH. In order to show a statement applicable to a wider cases, I work in a general setting. I appreciate any corrections if you find errors. = Convention and Terminology = Let \((X,<)\) denote a strictly partially ordered set throughout this blog post. A map \begin{eqnarray*} [ \ ] \colon X \times \mathbb{N} & \to & X \\ (x,n) & \mapsto & xn \end{eqnarray*} is said to be a system of fundamental sequences on \((X,<)\) if it satisfies the following: # For any \(x \in X\), if \(x0 = x\), then the following hold: ## \(x\) is minimal with respect to \(<\). ## \(xn = x\) for any \(n \in \mathbb{N}\). # For any \(x \in X\), if \(x0 \neq x\), then the following hold: ## For any \(n \in \mathbb{N}\), \(xn < x\). ## If \(x1 = x0\), then the following hold: ### For any \(y \in X\), if \(y < x\), then \(y = x0\) or \(y < x0\). ### For any \(n \in \mathbb{N}\), \(xn = x0\). ## If \(x1 \neq x0\), then the following hold: ### For any \(y \in X\), if \(y < x\), then there exists an \(n \in \mathbb{N}\) such that \(y < xn\). ### For any \(n \in \mathbb{N}\), \(xn < xn+1\). Obviously, the notion of a system of fundamental sequences of a strictly partially ordered set is a generalisation of those for countable ordinals and ordinal notations. We say that a system \([ \ ]\) of fundamental sequences of \((X,<)\) is well-founded if for any pair \((x,n)\) of an \(x \in X\) and an \((n_i)_{i=0}^{\infty} \in \mathbb{N}^{\omega}\), if \(n\) is primitive recursive, then there exists a pair \((k,y)\) of a \(k \in \mathbb{N}\) and an \(y = (y_i)_{i=0}^{k} \in X^{k+1}\) satisfying the following: # \(y_0 = x\). # For any \(i \in \mathbb{N}\) smaller than \(k\), \(y_{i+1} = y_in_i\). # \(y_k0 = y_k\). The condition that \(n\) is primitive recursive is often assumed so that the well-foundedness is formalisable in arithmetic. If you are interested only in set theory, then you can drop the condition. In this case, the resulting well-foundedness will get just stronger, and hence the replacement does not matter in the arguments below. = FGH and Lemmata = Let \([ \ ]\) be a well-founded system of fundamental sequences throughout this section. Put \(S := ((X,<),[ \ ])\). I define a map \begin{eqnarray*} f_{S,\bullet} \colon X \times \mathbb{N} & \to & \mathbb{N} \\ (x,n) & \mapsto & f_{S,x}(n) \end{eqnarray*} by the following transfinite recursion along \(<\): # If \(x0 = x\), then \(f_{S,x}(n) = n+1\). # If \(x0 \neq x\) and \(x1 = x0\), then \(f_{S,x}(n) = f_{S,x0}^n(n)\). # If \(x0 \neq x\) and \(x1 \neq x0\), then \(f_{S,x}(n) = f_{S,xn}(n)\). Obviously, it is a generalisation of the FGHs associated to systems of fundamental sequences for ordinals and ordinal notations. The well-definedness of \(f_S\) is guaranteed by the well-foundedness of \([ \ ]\), and hence this construction does not work for a general notation without the well-foundedness. Similar constructions work for HH and SGH. We say that \(f_S\) is order-preserving if for any \((x,y) \in X^2\), \(x < y\) implies \(f_{S,x} < f_{S,y}\), where the latter order is defined as the eventual domination. It is known that the FGHs associated to systems of fundamental sequences for ordinals and ordinal notations are not necessarily order-preserving. For example, I constructed an example of a system \([ \ ]\) of fundamental sequences for ordinals below \(\omega^{\omega}+1\) satisfying \(f_{((\omega^{\omega}+1,\in),[ \ ]),\omega^{\omega})} < f_{((\omega^{\omega}+1,\in),[ \ ]),\omega+1)}\), and Professor Kihara constructed an example of a system \([ \ ]\) of fundamental sequences for ordinals below \(\omega_1^{\textrm{CK}}+1\) induced by the canonical system of fundamental sequences on Kleene's \(\mathcal{O}\) associated to an enumeration of Turing machines satisfying \(f_{((\omega_1^{\textrm{CK}}+1,\in),[ \ ]),\omega_1^{\textrm{CK}})} < f_{((\omega_1^{\textrm{CK}}+1,\in),[ \ ]),\omega+3)}\). Now I prepare two useful lemmata, whose counterparts for HH and SGH hold by similar arguments. ; Proof. : It immediately follows by transfinite induction on \(x\) along \(<\). : □ ; Proof. : Let \(y \in X\). Suppose \(x < y\). We show that \(n > N\) implies \(f_{S,x}(n) < f_{S,y}(n)\) for any \(n \in \mathbb{N}\) by transfinite induction on \(y\) along \(<\). By the condition of \(N\), we have \(x < yN\). Since \([ \ ]\) is a system of fundamental sequence, we have \(yN = yn\) or \(yN < yn\). Therefore we obtain \(x < yn\). By \(x < y\), \(y\) is not minimal, and hence \(y0 \neq y\). Therefore \(f_{S,y}(n)\) coincides with either \(f_{S,yn}(n)\) or \(f_{S,yn}^n(n)\). It implies \(f_{S,yn}(n) \leq f_{S,y}(n)\) by \(n > N \geq 0\) and the elementary inequality. : By induction hypothesis, \(m > N\) implies \(f_{S,x}(m) < f_{S,yn}(m)\) for any \(m \in \mathbb{N}\). Applying it to the case \(m = n\), we obtain \(f_{S,x}(n) < f_{S,yn}(n) \leq f_{S,y}(n)\). : □ = Syst3ms' Condition and Main Result = A system \([ \ ]\) of fundamental sequences on \((X,<)\) is said to be a syst3m of fundamental sequences on \((X,<)\) if it satisfies the following: # \([ \ ]\) is well-founded. # For any \(x \in X\), there exists an \(N \in \mathbb{N}\) such that for any \(y \in X\), \(x < y\) implies \(x < yN\). We denote by \(N_x \in \mathbb{N}\) the minimum of such an \(N\). The condition holds in various settings. In many cases where \((X,<)\) is an ordinal notation coded as a set of formal strings and \([ \ ]\) is given by a typical nesting rule, \(N\) can be taken as the length of \(x\) as formal strings plus \(1\). I should clarify that I am not stating that this condition holds for any notations which you know, in order to avoid people who do not read precise conditions to state that FGHs associated to their notations are order-preserving by my argument. Let \([ \ ]\) denote a syst3m of fundamental sequences on \((X,<)\) thoughout this blog post from now on. Put \(S := ((X,<),[ \ ])\). Now we are ready for the main theorem. ; Proof. : The assertion immediately follows from the fundamental lemma. : □ Since the counterparts of the fundamental lemma hold also for HH and SGH, the counterparts of the order-preserving property of FGH can be verified in the same way for them. In particular, we obtain the following corollary: ; Proof. : The assertion immediately follows from the counterpart of the order-preserving property of FGH for HH, because the expansion yields a syst3m of fundamental sequences under the assumption. : □ = Open Problem and Alternative Solution = If FGH along a specific system of fundamental sequences does not satisfy the order-preserving property, then the resulting functions are not necessarily strictly increasing. Then a natural question arises: The statement is easily verified in the case where \(x\) is minimal or a successor such that \(x0\) satisfies the statement. On the other hand, in order to deal with the case where \(x\) is a limit, then we need to refer to \(N_{xn}\)'s if we want to apply the fundamental lemma. Moreover, in order to apply transfinite induction on \(x\) along \(<\), we need to refer to all elements \(y\) of \(X\) obtained by repeating to apply \([ \ ]\) to \(x\) in a primitive recursive way. Therefore if the set of \(N_y\)'s for such \(y\)'s is bounded, then the statement is true. However, such a condition rarely holds. Therefore the eventually strict increasing property of FGH is still open. I note that even if we do not know whether a given syst3m of fundamental sequences satisfies the eventually strictly increasing property, we can easily modify FGH so that it actually outputs desired large numbers. For example, I define a map \begin{eqnarray*} F_{S,\bullet} \colon X \times \mathbb{N} & \to & \mathbb{N} \\ (x,n) & \mapsto & F_{S,x}(n) \end{eqnarray*} by the following transfinite recursion along \(<\): # If \(x0 = x\), then \(F_{S,x}(n) = n+1\). # If \(x0 \neq x\) and \(x1 = x0\), then \(F_{S,x}(n) = F_{S,x0}^n(n)\). # If \(x0 \neq x\) and \(x1 \neq x0\), then \(F_{S,x}(n) = \sum_{m = 0}^{n} F_{S,xm}(m)\). Then it is easy to see that \(F_{S,x}\) is strictly increasing for any \(x \in X\), and is greater than \(f_{S,x}\). Of course, the counterpart of the fundamental lemma can be verified by a completely similar argument, and hence the counterpart of the order-preserving property of FGH also holds by the same argument. Therefore it might be better to use such a variant in order to ensure that our notations actually yield desired large numbers, although it does nothing significant if the original FGH actually satisfies the strictly increasing property. One issue is that if we employ it, then people without understanding the problem on FGH might give us comments like "Why are you using such a saladible extension of FGH? Unfortunately, it is quite trivially well-known by experts, or even by standard googologists, that such a replacement never changes the growth rate at all. I WIN!!!!" Category:Blog posts